The Official Records compiled by the US Government of both Union and Confederate armies called it the War of the Rebellion. The Civil War is what people chose to call it.
As to the question of what the definition of a Civil War is, the Geneva Convention offers the following criteria.
1. The party in revolt must be in possession of a part of the national territory.
2. The insurgent civil authority must exercise de facto authority over the population within the determinate portion of the national territory.
3. The insurgents must have some amount of recognition as a belligerent.
4. The legal Government is "obliged to have recourse to the regular military forces against insurgents organized as military."
The Confederacy meet those criteria, so calling it a Civil War is not wrong.
Thanks for the info!